APSIDAL ANGLES FOR SYMMETRICAL DYNAMICAL
SYSTEMS WITH TWO DEGREES OF FREEDOM
J. L. SYNGE
The bob of a spherical pendulum (or particle on a smooth sphere
under gravity) oscillates between two levels, and the change of
azimuth in passing from the lower to the higher level, or vice versa,
is the same for all such passages during a single motion. Using the
language of orbit theory, we shall refer to this change of azimuth as
the apsidal angle. It is a function of the two constants of the motion,
total energy and angular momentum, but the remarkable fact is
that the apsidal angle a always satisfies the inequalities1
(1)
T/2
<a < T.
The purpose of the present note is not to add to the theory of the
spherical pendulum, but rather to place the problem in a more
general setting. Consider a particle of unit mass which moves on a
surface of revolution 2 under the action of a conservative field of
force for which the potential energy is independent of azimuth.
Let A be the axis of 2. Then S is determined completely by the
section C of 2 by a half-plane terminated by A. Since the particle
cannot pass from a portion of S to a disconnected portion of 2, we
may suppose C to be a single connected curve.
Several cases present themselves:
(i) C is an open curve with both ends on A (2 homeomorphic to a
sphere).
(ii) C is an open curve with one end on A and the other end at
infinity (2 homeomorphic to a paraboloid of revolution).
(iii) C is an open curve with both ends at infinity (2 homeomorphic
to a cylinder).
(iv) C is a closed curve which does not meet A (2 homeomorphic
to a torus).
In any one of these cases, we assign coordinates Rt <£, where R is
arc length measured along C and <t> is the azimuthal angle. The range
of 4> is taken to be (— oo, oo), an increase of 2TT in <£ leading us back
to the same point. The range of R depends on the type of C. In case
(i), the range of R is finite; in case (ii), the range may be taken as
Presented to the Society, April 26, 1947; received by the editors April 12, 1947.
1
For references to earlier work, see P. Appell, Traité de mécanique rationnelle,
vol. 1 (1926) p. 518; see also A. Weinstein, Amer. Math. Monthly vol. 49 (1942) pp.
521-523, W. Kohn, Trans. Amer. Math. Soc. vol. 59 (1946) pp. 107-131.
Ill
112
J. L. SYNGE
[February
(0, oo); in case (iii), the range is (— oo, GO); in case (iv), the range
is taken to be ( - c o , oo), an increase in R equal to the length of C
leading us back to the same point.
The line element of 2 may be written
da2 = dR2 + B{R)d<t>\
(2)
and the potential energy is V(R). At any point common to 2 and the
axis A we have B(R) = 0. In case (iv) B and V are periodic functions
of R, the period being the length of C We shall assume B and V to be
of class C2 in all domains where the motion of the particle is considered.
Although the dynamical system described above consists of a
particle on a surface of revolution, it is clear that any results obtained
will be available also for a conservative dynamical system with two
degrees of freedom and one ignorable coordinate, the Lagrangian
function being
L = 2^[A(R)R2
(3)
+ B(R)4>*] -
V(R).
We shall, however, continue to refer to the system as if it consisted
of a particle on a surface of revolution.
According to Jacobi's principle of stationary action, the path of the
particle, if moving with total energy £ , is a geodesic in a two-space
with metric ds where
ds2 = (E -
(4)
V)da2.
Let us introduce instead of R a new coordinate r, defined by
ƒ«
the lower limit of integration being arbitrary. Then (4) becomes, by
use of (2) and (5),
ds2 = dr2 + G(r)d<l>2,
(6)
where
(7)
G(r) -
[E-V{R)]B(R).
The form (6) may itself be regarded as the line element of a surface
of revolution 5. Accordingly the study of particle paths on a surface
of revolution S reduces to the study of geodesies on another surface of
revolution S.2
2
In general, S cannot be imbedded in a Euclidean 3-space. Since we shall be concerned solely with the intrinsic properties of S, that fact is not important for the purposes of the present paper.
1948]
APSIDAL ANGLES FOR SYMMETRICAL DYNAMICAL SYSTEMS
113
The geodesies for (6) satisfy the Lagrangian equations
d
dw
ds dr'
dw
d
dw
dr ~~ '
ds d<f>'
dw
d<f> ~"
'
where the prime means d/ds and
w = 2-l[r"+G(r)*'f].
(9)
Since <j> is ignorable, we have the first integral
(io)
Gif)*' = *,
a constant analogous to the constant of angular momentum. Equation (10) replaces the second of (8) ; instead of the first of (8), we may
use the first integral
(11)
r / 2 + G ( r ) « , 2 = 1.
Elimination of 5 between (10) and (11) gives
(12)
(dr/dj)* « G(G - h*)/h*.
The two constants, E and hf are involved in the theory, but they
play very different roles. The constant E is implicit in the line element
(6), through G, but the line element does not involve h. Since
(13)
E = T + V,
where T is the kinetic energy, and T cannot be negative, we see
that E is greater than or equal to V at any point in the path of the
particle. This places on E the condition
(14)
E §: minimum of V on 2).
The constant h must satisfy a more exacting inequality. Since r ' 2 ^ 0
and G(r)^0 by (7) and the positive definite character of (2),
(15)
Â2 g G(r)
follows from (10) and (11), for all values of r on the particle path;
this may also be seen directly from (12).
The apsides of a particle path correspond to maximum and minimum values of R, and the apsidal angle a is the increment in the
azimuthal angle <j> in passing from minimum to maximum or from
maximum to minimum. 3 Since the transformation (5) is monotone,
8
The surface of revolution is a particular case of the radial manifold, and radial
apsides coincide with potential apsides; cf. J. L. Synge, Trans. Amer. Math. Soc. vol.
34 (1932) pp. 481-522.
114
J. L. SYNGE
[February
and <f> is not transformed in the passage from 2 to S, it follows that
the apsidal distances (i?i, R2) for the particle path on 2 correspond by
(5) to the apsidal distances (n., r*) for the corresponding geodesic on S,
and the apsidal angles are equal.
T o find the apsidal distances, we put dr/d<f> = 0 in (12). We have
then to consider the two equations G = 0 and G — &2 = 0. By (10),
G = 0 implies h = 0, and so G = 0 implies G — &2 = 0. Consequently we
have to consider only the equation
(16)
G = h\
However, the case h = 0 is singular, and we shall exclude it from
further consideration. By (10), it implies 0 ' = O for values of r not
making G = 0, and so the motion is along a meridian curve (simple
pendulum motion for a spherical pendulum).
If (16) has no roots, there are no apsides. To study the existence of
apsides, let us regard E as assigned, so that the function G(r) is determined. Oscillation between apsidal distances n , H (n <r*) will occur
if, and only if, there exists a pair of roots of (16) :
(17)
G(ri) = h\
G(n) = h>
(n < r 8 ),
with the inequality (IS) satisfied for all intermediate values of r:
G(r) ^ h2
(18)
( f i ^ r g r%).
The scheme for finding apsidal distances is then clear. The constant
E being assigned, we consider the graph of G(r) versus r. Unless the
graph shows a t least one relative maximum, a pair of apsidal distances cannot occur. Let us suppose that r = r0 gives a relative maximum, so that the derivative vanishes:
(19)
(dG/dr)r„ro - 0.
We then draw a parallel to the r-axis below this maximum, cutting
the curve of G{r) at n and a t r2, with ri<ro<r2l and assign to h the
value given by (17). Then there exists a geodesic with apsidal distances r lf r2, and, by (12), the apsidal angle is given by
(20)
a = f
f
"à* = h f ' [G(G - Â 2 )]- 1 ' 2 ^.
It is interesting to note that if we had chosen h2 = G(r0) we would
have obtained a circular geodesic (corresponding to the conical
pendulum). This is easily seen from equations (8)-(12). In fact,
any geodesic on S possessing two apsidal distances weaves in and out
across a circular geodesic.
1948]
APSIDAL ANGLES FOR SYMMETRICAL DYNAMICAL SYSTEMS
115
Let us now consider bounds for the apsidal angle a, assuming that
in the range (n, r2) there is only one value ro satisfying (19). Then in
each of the ranges (ri, r0), (r0, r2) G is a monotone function of r.
We may write (20) in the form
» fOSfQ
/
(21)
[G(G~ W)]~u*\dr/dG\dG
+ h f ' r° [G(G - A2)]-1'21 dr/dG | dG
= h f
°[G(G - A2)]-1/2!! dr/dG|! + | dr/dG|2]dG,
where G0 = G(r0), and the subscripts 1 and 2 refer respectively to the
ranges (n, r0) and (r0, r2).
Consider now the ratio
, x
(22)
IdG/drl
1' 2
'
(Go-G)
'
in the range (ri, r2). As r approaches r0, this ratio has the limit
( - 2d2G/dr2)1'2,
evaluated at r = r0. Accordingly the ratio (22) is bounded above and
below, and we may write
(23)
m S (Go - G)-1/21 dG/dr \£M
(rx ^ r £ r«)f
where m and If are in general functions of E and h. Equivalently,
(24)
Af-i(Go - G)-i/t g | dr/dG | £ «r^Go - G)"1'2
(ri S f S r*)-
We might have m = 0, in which case the upper bound is infinite. With
the inequalities (24), equation (21) gives
M- 1 / SaS
(25)
nrxIt
where
(26)
/ = 2k f
°[G(Go -G)(G-
h^Y^dG.
Changing the variable of integration by the transformation
(27)
we get
G = h* cos2 6 + Go sin2 6,
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J. L. SYNGE
> T/2
ƒo
[February
[1 + (Go/A2 - 1) sin2 e]-l'He,
and hence, since G 0 >A 2 ,
(29)
2TTÂG71/2 < I < 2TT.
Substitution in (25) gives the following result: For motion on a surface
of revolution S (or, equivalentlyy for a geodesic on a surface of revolution
S with line element (6)), the apsidal angle a satisfies the inequalities
(30)
27rM~1^G71/2 < a < lirnT1,
where h is the constant of the first integral (10), Go is G(r0) where r0
satisfies (19), and M, m are respectively upper and lower bounds of the
ratio (22) in the range of oscillation (ri, r 2 ), determined by G = h2.
It is interesting to see what happens when the geodesic under
consideration lies close to the circular geodesic r = r0. Then h2 approaches Go and the two bounds M and m approach the common
value ( — 2d2G/dr2)~~1/2 evaluated at r 0 ; we get in the limit
(31)
a = 2T(-
2d2G/dr2)-li2,
evaluated at r0. This value is easily checked by differentiating (12),
dividing by dr/dxf), and linearizing the resulting equation.
Having transformed the problem of the apsidal angle for motion
on a surface of revolution to that for geodesies on a surface of revolution, we can bring to bear on it well known results in differential
geometry. Consider, on the surface of revolution 5, the triangle AOB
the sides of which are geodesies. Let O be the pole of 5, that is, the
point where it is met by the axis. Then OA and OB are meridians.
Further, suppose that A and B are adjacent apsides on the geodesic
AB, so that the angles ABO and BAO are right angles and AOB is
the apsidal angle a. Then a is the excess of the angle sum over 7r,
and so by the well known formula
(32)
a = f f
KdS,
where K is the Gaussian curvature of S for the metric (6) and the
integration extends over the triangle AOB, dS being the element of
area.
The formula (32) brings an interesting fact to light. Consider
the small oscillations of a spherical pendulum about the position of
equilibrium. It is known that the particle path is approximately a
i 9 48]
APSIDAL ANGLES FOR SYMMETRICAL DYNAMICAL SYSTEMS
117
central ellipse (a = 7r/2). Consider the triangle AOB as described
above. To that triangle on S there corresponds on the sphere 2 a
triangle which we may also call AOB; the sides A0 and OB are great
circles joining 0 , the lowest point of the sphere 2 , to adjacent apsides
A and B\ AB is the particle path between apsides.
Now, for small oscillations, the area of the triangle AOB on 2 is
small. The area of AOB on S is still smaller, since the factor E—V
involved in passing from da2 to ds2 is small for small kinetic energy.
How then can the integral in (32) have a finite value, approximately
7r/2? The answer is to be found in K. For the metric (6), the circle
V=E is singular, and the Gaussian curvature of 5 tends to infinity
as we approach this circle. If we take the zero of potential energy
at the lowest point of the sphere, that is, at the point 0, then small
oscillations correspond to small values of E, and the radius of the
circle V=E is small. But inside this circle, considered in S, the
Gaussian curvature is great, in such a way that its greatness balances
the smallness of the area of the circle.
These considerations point to the desirability of the further investigation of the geometry of dynamical systems when the Jacobi
metric (6) is used; this remark applies not merely to the simple
symmetric case discussed, above, but to a general dynamical system.
The locus V=E has an interesting geometry. It is a null-domain, in
the sense that the length of any curve drawn in it is zero. It forms a
barrier across which a geodesic cannot pass; in fact, geodesies approaching it are bent back sharply. Furthermore, on it the curvature
becomes infinite.
It might seem that the reduction of the question of dynamical
apsidal angles to the question of geodesic apsidal angles might yield
some simple general bounds. So we ask: For the whole class of surfaces of revolution, do there exist upper and lower bounds for apsidal
angles? We shall show that the answer is in the negative, by constructing models in which the apsidal angles pass any given bound.
It is simplest to use limiting models with sharp edges; passage to the
limit from smooth surfaces will not destroy the inequalities.
Consider a long circular cylinder with flat ends, the pole 0 being
the center of one end E. Consider a geodesic which consists in part
of a chord of E, not a diameter. The midpoint of this chord is an
apse; in fact, it is a point of minimum distance from 0. Pursuing the
geodesic in one direction, it passes over the edge of £ , and winds
round and round the curved sides of the cylinder. The azimuthal
angle increases by 2w in each such revolution, and the next apse cannot occur until the geodesic reaches the other end of the cylinder.
118
J. L. SYNGE
Obviously, by making the cylinder sufficiently long, we can make the
apsidal angle as great as we please.
To get a small apsidal angle, we take as our surface the two faces
of a flat circular disk. The pole 0 is the center of one face E. Consider
a geodesic consisting in part of a chord of E which subtends at 0 a
small angle j8. The midpoint of the chord is an apse. Follow this
geodesic in one direction over the edge of the disk and on to the other
face E'. The chord on E' and the chord on E will make equal angles
with the radius drawn from O to the point where the geodesic passes
over the edge. The midpoint of the chord on E1 will also be an apse.
The apsidal angle is a=j8, and can be made as small as we please
by making ]3 sufficiently small.
Consequently, for the whole class of surfaces of revolution, there
is neither an upper nor a lower bound for the geodesic apsidal angle.
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